MathDB
Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1980 Vietnam National Olympiad
1980 Vietnam National Olympiad
Part of
Vietnam National Olympiad
Subcontests
(3)
3
2
Hide problems
Midpoints of segments forming triangles with equal areas.
Let
P
P
P
be a point inside a triangle
A
1
A
2
A
3
A_1A_2A_3
A
1
A
2
A
3
. For
i
=
1
,
2
,
3
i = 1, 2, 3
i
=
1
,
2
,
3
, line
P
A
i
PA_i
P
A
i
intersects the side opposite to
A
i
A_i
A
i
at
B
i
B_i
B
i
. Let
C
i
C_i
C
i
and
D
i
D_i
D
i
be the midpoints of
A
i
B
i
A_iB_i
A
i
B
i
and
P
B
i
PB_i
P
B
i
, respectively. Prove that the areas of the triangles
C
1
C
2
C
3
C_1C_2C_3
C
1
C
2
C
3
and
D
1
D
2
D
3
D_1D_2D_3
D
1
D
2
D
3
are equal.
Maximum of the sum x_ix_{i+1} in terms of p, the sum of x_i.
Let be given an integer n\ge 2 and a positive real number
p
p
p
. Find the maximum of
∑
i
=
1
n
−
1
x
i
x
i
+
1
,
\displaystyle\sum_{i=1}^{n-1} x_ix_{i+1},
i
=
1
∑
n
−
1
x
i
x
i
+
1
,
where
x
i
x_i
x
i
are non-negative real numbers with sum
p
p
p
.
2
2
Hide problems
Sum (m_i+1/m_i)^2>=k(k/S+S/k) where Sum m_i=S.
Let
m
1
,
m
2
,
⋯
,
m
k
m_1, m_2, \cdots ,m_k
m
1
,
m
2
,
⋯
,
m
k
be positive numbers with the sum
S
S
S
. Prove that \displaystyle\sum_{i=1}^k\left(m_i +\frac{1}{m_i}\right)^2 \ge k\left(\frac{k}{S}+\frac{S}{k}\right)^2
Can x^3-2x^2-2x+m = 0 have three different rational roots?
Can the equation
x
3
−
2
x
2
−
2
x
+
m
=
0
x^3-2x^2-2x+m = 0
x
3
−
2
x
2
−
2
x
+
m
=
0
have three different rational roots?
1
2
Hide problems
Sum of sines is greater than 1.
Let \alpha_{1}, \alpha_{2}, \cdots , \alpha_{ n} be numbers in the interval [0, 2\pi] such that the number \displaystyle\sum_{i=1}^n (1 + \cos \alpha_{ i}) is an odd integer. Prove that \displaystyle\sum_{i=1}^n \sin \alpha_i \ge 1
Ratio of projections of a tetrahedron on two planes.
Prove that for any tetrahedron in space, it is possible to find two perpendicular planes such that ratio between the projections of the tetrahedron on the two planes lies in the interval
[
1
2
,
2
]
.
[\frac{1}{\sqrt{2}}, \sqrt{2}].
[
2
1
,
2
]
.